mKdV ZK Equation Research Articles

Newly formed solitary wave solutions and other solitons to the (3+1)-dimensional mKdV–ZK equation utilizing a new modified Sardar sub-equation approach

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the [Formula: see text]-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

In this research, we investigate the effects of fractional order on the (3 + 1)-dimensional generalized space-time fractional modified KdV-Zakharov-Kuznetsov (mKdV-ZK) equation. We approach the problem by utilizing the conformable fractional derivative. By reducing the mKdV-ZK equation to an integer order nonlinear ordinary differential equation, we apply the Jacobi elliptic function method to find exact solutions. These solutions are specifically tailored for the fractional order of the (3 + 1)-dimensional generalized mKdV-ZK equation, encompassing solitary waves, shock waves, and periodic waves. We also compare these exact solutions with fractional solutions to gain further insights. Notably, our approach demonstrates the feasibility of solving nonlinear time-fractional differential equations with conformable derivatives. Several diagrams have been included to visually depict the behavior of the solutions under fractional order when certain special parameter values are employed.

Traveling Wave Solutions for Time-Fractional mKdV-ZK Equation of Weakly Nonlinear Ion-Acoustic Waves in Magnetized Electron–Positron Plasma

In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on the properties of certain employed truncated M-fractional derivatives, we reduce the time-fractional mKdV-ZK equation to an integer-order ordinary differential equation utilizing an adequate traveling wave transformation. Further, we derive a dynamical system to present bifurcation of the equation equilibria and show existence of solitary and kink singular wave solutions for the time-fractional mKdV-ZK equation. Furthermore, we establish symmetric solitary, kink, and singular wave solutions for the governing model by using the ansatz method. Moreover, we depict desired results at different physical parameter values to provide physical interpolations for the aforementioned equation. Finally, we introduce applications of the governing model in detail.

Bifurcation of Travelling Wave Solutions for (3+1)-dimensional mKdV-ZK Equation

Since the (3+1)-dimensional modified Korteweg de Vries- Zakharov–Kuznetsov (mKdV-ZK equation) was proposed, many methods have been used to find its exact solution, but some solutions are always missed in the calculation process. In this paper, the mKdV-ZK equation is studied by the planar dynamical systems method, the bifurcation diagram and phase diagram under different parameters are obtained, and the solution of the equation is numerically simulated. Finally, the traveling wave solutions of the mKdV-ZK equation is constructed under different parameters.

Computational Technique to Study Analytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation

In this article, we study and investigate the analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation. We have got new exact solutions of the fractional mKDV-ZK equation by using first integral method; we found new types of hyperbolic solutions and trigonometric solutions by symbolic computation.

The (3+1)-dimensional generalized mKdV-ZK equation for ion-acoustic waves in quantum plasmas as well as its non-resonant multiwave solution* *Project supported by the National Natural Science Foundation of China (Grant No. 11975143), the Natural Science Foundation of Shandong Province of China (Grant No. ZR2018MA017), the Taishan Scholars Program of Shandong Province, China (Grant No. ts20190936), and the Shandong University of Science

The quantum hydrodynamic model for ion-acoustic waves in plasmas is studied. First, we design a new disturbance expansion to describe the ion fluid velocity and electric field potential. It should be emphasized that the piecewise function perturbation form is new with great difference from the previous perturbation. Then, based on the piecewise function perturbation, a (3+1)-dimensional generalized modified Korteweg–de Vries Zakharov–Kuznetsov (mKdV-ZK) equation is derived for the first time, which is an extended form of the classical mKdV equation and the ZK equation. The (3+1)-dimensional generalized time-space fractional mKdV-ZK equation is constructed using the semi-inverse method and the fractional variational principle. Obviously, it is more accurate to depict some complex plasma processes and phenomena. Further, the conservation laws of the generalized time-space fractional mKdV-ZK equation are discussed. Finally, using the multi-exponential function method, the non-resonant multiwave solutions are constructed, and the characteristics of ion-acoustic waves are well described.

Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Abstract In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given.

A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations

AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville and the Riccati-Bernoulli Sub-ODE method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The Riccati-Bernoulli Sub-ODE method appears to be easier and more convenient by means of a symbolic computation system.

Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative

Fractional order nonlinear evolution equations involving conformable fractional derivative are formulated and revealed for attractive solutions to depict the physical phenomena of nonlinear mechanisms in the real world. The core aim of this article is to explore further new general exact traveling wave solutions of nonlinear fractional evolution equations, namely, the space time fractional (2+1)-dimensional dispersive long wave equations, the (3+1)-dimensional space time fractional mKdV-ZK equation and the space time fractional modified regularized long-wave equation. The mentioned equations are firstly turned into the fractional order ordinary differential equations with the aid of a suitable composite transformation and then hunted their solutions by means of recently established fractional generalized ($D_\xi^\alpha G/G$)-expansion method. This productive method successfully generates many new and general closed form traveling wave solutions in accurate, reliable and efficient way in terms of hyperbolic, trigonometric and rational. The obtained results might play important roles for describing the complex phenomena related to science and engineering and also be newly recorded in the literature for their high acceptance. The suggested method will draw the attention to the researchers to establish further new solutions to any other nonlinear evolution equations.

Analysis of Lie symmetries with conservation laws for the (3+1) dimensional time-fractional mKdV–ZK equation in ion-acoustic waves

In this paper, symmetry properties and conservation laws of the (3+1) dimensional time-fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation have been studied with Riemann–Liouville fractional derivative. Fractional Lie symmetry method has been used here for getting symmetry properties of (3+1) dimensional time-fractional mKdV–ZK equation. Here, the reduction of (3+1) dimensional time-fractional mKdV–ZK equation into fractional ordinary differential equation has been done by using Erdelyi–Kober fractional differential operator. Also, by using the new conservation theorem, the new conserved vectors for (3+1) dimensional time-fractional mKdV–ZK equation have been constructed with the help of formal Lagrangian with a detail derivation.

Lie symmetry analysis for similarity reduction and exact solutions of modified KdV–Zakharov–Kuznetsov equation

In this paper, the Lie group analysis is used to carry out the similarity reduction and exact solutions of the \((3+1)\)-dimensional modified KdV–Zakharov–Kuznetsov equation. This research deals with the similarity solutions of mKdV–ZK equation. The mKdV–ZK equation has been reduced into a new partial differential equation with less number of independent variables, and again using Lie group symmetry method, the new partial differential equation is reduced into an ordinary differential equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group, and similarity reduction for the mKdV–ZK equation. In addition to that, solitary wave solutions of the mKdV–ZK equation are derived from the reduction equation. Thus, we obtain some new exact explicit solutions of the \((3+1)\)-dimensional mKdV–ZK equation which describes the dynamics of nonlinear waves in plasmas.

Stability analysis solutions for nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a magnetized electron–positron plasma

The nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov ​(mKdV–ZK) equation governs the behavior of weakly nonlinear ion-acoustic waves in magnetized electron–positron plasma which consists of equal hot and cool components of each species. By using the reductive perturbation procedure leads to a mKdV–ZK equation governing the oblique propagation of nonlinear electrostatic modes. The stability of solitary traveling wave solutions of the mKdV–ZK equation to three-dimensional long-wavelength perturbations is investigated. We found the electrostatic field potential and electric field in form traveling wave solutions for three-dimensional mKdV–ZK equation. The solutions for the mKdV–ZK equation are obtained precisely and efficiency of the method can be demonstrated.

Exact traveling wave solutions to the (3+1)-dimensional mKdV–ZK and the (2+1)-dimensional Burgers equations via exp(−Φ(η))-expansion method

In this work, the exact traveling wave solutions to the (3+1)-dimensional mKdV–ZK equation and the (2+1)-dimensional Burgers equation are studied using the exp(-Φ(η))-expansion method. The traveling wave solutions are expressed in terms of the exponential functions, the hyperbolic functions, the trigonometric functions and the rational functions. This method is one of the powerful methods that appear in recent time in establishing some new exact traveling wave solutions to the nonlinear partial differential equations. It is shown that the exp(-Φ(η))-expansion method is simple and valuable mathematical instrument for solving nonlinear evolution equations in mathematical physics and engineering.

Exact travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation using the new approach of generalized G ′ / G $\left (\boldsymbol { {G^{\prime }/G}} \right )$ -expansion method

In this paper, the new generalized ( $G^{\prime } /G)$ -expansion method is executed to find the travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation. The efficiency of this method for finding exact and travelling wave solutions has been demonstrated. It is shown that the new approach of generalized ( ${G}^{\prime }/G)$ -expansion method is a straightforward and effective mathematical tool for solving nonlinear evolution equations in applied mathematics, mathematical physics and engineering. Moreover, this procedure reduces the large volume of calculations.